STEPPED COLUMNS

Stepped columns are often used in mill buildings to support roof or upper wall loads and runway girders on which cranes travel, as illustrated in Fig. 3.56.

A notional load approach to the design of mill building columns has been pre-sented by Schmidt (2001). In general, however, the effective length of a uniform or prismatic column having the same buckling characteristics as that of the stepped

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FIGURE 3.55 Column with perforated web plates.

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FIGURE 3.56 Type of columns in mill buildings.

column is needed to determine the axial compressive resistance and the Euler buck-ling load for potential modes of failure by buckbuck-ling or bending about the x–x axis (see Fig. 3.57). Tables to determine effective-length factors are provided in the Association of Iron and Steel Engineers guide (AISE, 1979) for a range of the three parameters defined in Fig. 3.57 and for the cases when the column base is either fixed or pinned and the column top is pinned. The parameter a is the ratio of the length of the upper (reduced) segment to the total length. The parameter B is the ratio of the second moment of area (about the centroidal x–x axes) of the combined (lower) column cross section to that of the upper section. The parameter P1/P2 is the ratio of the axial force acting in the upper segment (roof and upper wall loads) to that applied to the lower segment (crane girder reactions with an allowance for lower wall loads and the column weight). Other notations are also given in Fig. 3.57. The AISE tables give ranges for a from 0.10 to 0.50, for B from 1.0 to 100, and for P1/P2 from 0.0 to 0.25. Huang (1968) provides values of the

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FIGURE 3.57 Notation for stepped columns (AISE, 1979).

effective-length factors in graphs over a somewhat different range of parameters for stepped columns with fixed bases and pinned tops and for values of P1/P₂ up to 1.0. Further values tabulated for the effective-length factors for stepped columns are given in Timoshenko and Gere (1961), the Column Research Committee of Japan handbook (CRCJ, 1971, English edition), and Young (1989). A method for determining the effective-length factor of a stepped column without using charts was proposed by Lui and Sun (1995). A hand calculation method for determining the buckling load of a stepped column was also proposed by Fraser and Bridge (1990).

The interaction equations used for the design of stepped columns depend on the potential modes of failure and therefore on how the columns are braced. Stepped columns are usually laterally unsupported over the entire length for buckling or bending about the x–x axis. For buckling about the y–y axis, lateral support is usually provided at the level of the crane runway girder seat, location B in Fig. 3.57.

Therefore, the following potential failure modes exist:

1. Bending of the overall column in-plane about the x–x axis. It is for this case that the equivalent length for the stepped column is needed in order that the compressive resistance and the Euler buckling load may be determined.

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2. Buckling about the y–y axis of the lower segment.

3. Yielding of the cross section of the lower segment.

4. Buckling about the y–y axis of the upper (reduced) segment of the column.

5. Yielding of the cross section of the upper segment.

The interaction equations for axial compression and biaxial bending are Pu

where Mux would include second-order effects, which could be modeled by B1x = Cmx

1− Pu/P_ex ≥ 1.0 (3.45c) with a similar term for B1y related to bending about the y–y axis, Muy.

It is conservatively assumed that the moments Muy due to eccentric crane girder reactions are all taken by the lower segment and therefore the moments about the y–y axis in the upper segment are zero.

Following the AISC LRFD procedures, failure modes 1, 2, and 3 are considered together, as are modes 4 and 5. When using the AISC LRFD equations for failure modes 1, 2, and 3 combined, the following values are suggested. The factored axial load, Pu, is P1+ P2, and Muxand Muyare the maximum modified factored moments within the length. Based on AISE (1979), Cmx is taken as 0.85 when all bents are under simultaneous wind load with sidesway. For crane load combinations, with only one bent under consideration, take Cmx equal to 0.95. Because it has been assumed that the lower segment takes all the moments about the y–y axis, Cmy

equals 0.6 when the base is pinned and 0.4 when the base is fixed about the y–y axis. The compressive resistance is the least of φPnxof the equivalent column and φP_nyof the bottom segment, taking, in this case, K = 0.8 if the base is considered to be fully fixed, and K = 1.0 if the base is considered to be pinned. The factored moment resistance φbMnx is based on the possibility of lateral–torsional buckling and φbMny is the full cross-section strength. This being the case, both factors B1x

and B_1y as given previously are limited to a value of’ 1.0 or greater. The value of Pex is as determined for the equivalent column and Pey is based on 0.8 of the lower segment for a fixed base and 1.0 of the lower segment if the base is pinned.

When using the AISC LRFD equations for failure modes 4 and 5 combined for the upper segment, the values suggested parallel those for the lower segment with the exceptions that there are moments only about the x–x axis and Cmx depends directly on the shape of the moment diagram.

The five failure modes could also be checked independently when the values for checking are as follows: For bending failure about the x–x axis, failure mode 1 (the predominant moments are about the x–x axis), the factored axial load is P1+ P2, and Muxand Muyare the maximum modified factored moments within the

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length. Cmx is taken as 0.85 when all bents are under simultaneous wind load with sidesway. For crane load combinations, with only one bent under consideration, take Cmx equal to 0.95. Because it has been assumed that the lower segment resists all the moments about the y–y axis, Cmy = 0.6 when the base is pinned and 0.4 when the base is fixed about the y–y axis. The value of B1 is not restricted. The compressive resistance is the least of φP_nx of the equivalent column and φP_ny of the bottom segment with K = 0.8 if the base is taken to be fully fixed and with K = 1.0 if the base is taken as pinned. The factored moment resistances, φbMnx

and φbMny, are the full cross-section strengths. The value of Pex is as determined for the equivalent column and Pey is based on 0.8 of the lower segment for a fixed base and 1.0 of the lower segment if the base is pinned.

The values for checking the last four potential failure modes, modes 2 to 5, independently (for y–y axis buckling and for cross-section strength) are as follows.

The factored force effects Pu, Mux, and Muy are those for the appropriate segment.

When checking for cross-section strength, the resistances φPn, φbMnx, and φbMny

are the full factored cross-section strengths for the appropriate segment. When checking for y–y axis buckling, φPny is based on y–y axis end conditions for the segment being considered. The effective length is the full length AB for the upper segment and the length BC for the lower segment if full base fixity does not exist and 0.80 of the length BC if fully fixed at the base. In Eq. 3.45, Mnx

is the appropriate lateral–torsional buckling strength taking into account the shape of the moment diagram, and Mny is the full cross-section strength. Cmx and Cmy

have minimum values of 1.0 because the variation of moment has been accounted for in finding the moment resistance. The value of Pex is that determined for the equivalent column and Pey is taken as that for the appropriate segment.

A demonstration of the LRFD approach to the design of stepped crane columns has been presented by MacCrimmon and Kennedy (1997). The design of a column with constant cross section subject to lateral loads and moments applied at a bracket has been considered by Adams (1970). The upper and lower segments are treated as separate beam-columns. Horne and Ajmani (1971) and Albert et al. (1995) take into account the lateral support provided by the girts attached to one flange of the column.